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Q.
If $A = \begin{bmatrix}1&2&x\\ 3&-1&2\end{bmatrix} $ and $B= \begin{bmatrix}y\\ x\\ 1\end{bmatrix}$ be such that $AB = \begin{bmatrix}6\\ 8\end{bmatrix} $, then:
Matrices
Solution:
Let $A = \begin{bmatrix}1&2&x\\ 3&-1&2\end{bmatrix}$ and $B= \begin{bmatrix}y\\ x\\ 1\end{bmatrix} $
$AB = \begin{bmatrix}1&2&x\\ 3&-1&2\end{bmatrix} \begin{bmatrix}y\\ x\\ 1\end{bmatrix} $
$ \Rightarrow \begin{bmatrix}6\\ 8\end{bmatrix} = \begin{bmatrix}y+2x+x\\ 3y-x+2\end{bmatrix} $
$ \Rightarrow \begin{bmatrix}6\\ 8\end{bmatrix} = \begin{bmatrix}y+3x\\ 3y-x+2\end{bmatrix} $
$\Rightarrow \ y + 3x = 6$ and $3y - x = 6$
On solving, we get
$x = \frac{6}{5}$ and $y = \frac{12}{5}$
$\Rightarrow \ y = 2x $